Common Throughput Maximization in UAV-Enabled OFDMA Systems with Delay Consideration
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1
Common Throughput Maximization in
UAV-Enabled OFDMA Systems with Delay
Consideration
Qingqing Wu, Member, IEEE, and Rui Zhang, Fellow, IEEE
arXiv:1801.00444v2 [cs.IT] 6 Jan 2018
Abstract
The use of unmanned aerial vehicles (UAVs) as communication platforms is of great practical
significance in future wireless networks, especially for on-demand deployment in temporary events and
emergency situations. Although prior works have shown the performance improvement by exploiting the
UAV’s mobility, they mainly focus on delay-tolerant applications. As delay requirements fundamentally
limit the UAV’s mobility, it remains unknown whether the UAV is able to provide any performance gain
in delay-constrained communication scenarios. Motivated by this, we study in this paper a UAV-enabled
orthogonal frequency division multiple access (OFDMA) network where a UAV is dispatched as a mobile
base station (BS) to serve a group of users on the ground. We consider a minimum-rate ratio (MRR) for
each user, defined as the minimum instantaneous rate required over the average achievable throughput, to
flexibly adjust the percentage of its delay-constrained data traffic. Under a given set of constraints on the
users’ MRRs, we aim to maximize the minimum average throughput of all users by jointly optimizing the
UAV trajectory and OFDMA resource allocation. First, we show that the max-min throughput in general
decreases as the users’ MRR constraints become more stringent, which reveals a fundamental throughput-
delay tradeoff in UAV-enabled communications. Next, we propose an iterative parameter-assisted block
coordinate descent method to optimize the UAV trajectory and OFDMA resource allocation alternately,
by applying the successive convex optimization and the Lagrange duality, respectively. Furthermore, an
efficient and systematic UAV trajectory initialization scheme is proposed based on a simple circular
trajectory. Finally, simulation results are provided to verify our theoretical findings and demonstrate the
effectiveness of our proposed designs.
Index Terms
The authors are with the Department of Electrical and Computer Engineering, National University of Singapore,
email:{elewuqq, elezhang}@nus.edu.sg. Part of this work has been presented in IEEE APCC 2017 as an invited paper [1].2
UAV communications, delay constraint, throughput maximization, trajectory design, OFDMA, re-
source allocation.
I. I NTRODUCTION
As predicted in [2], the number of unmanned aerial vehicles (UAVs), also known as drones,
will continue to surge over the next several years, with the global annual unit shipment increasing
more than tenfold from 6.4 million in 2015 to 67.7 million by 2021. Such a dramatic increase is
due to the steadily decreasing cost of UAVs and their fast-growing demand in applications such as
surveillance and monitoring, aerial camera and radar, cargo delivery, communication platforms,
etc. In June 2017, the International Olympic Committee (IOC) and Intel announced to use UAVs
to enhance the future Olympic Games experience, such as the drone light show [3]. Just in the
same month, the “Safe DRONE Act of 2017” was proposed to the United States Congress for
accelerating the development of the UAV technology, which requested 14 million dollars for
funding a host of research projects on UAVs [4]. In fact, many prominent companies such as
Google, Qualcomm, Amazon, and Nokia, have already launched their respective programs to
advance the UAV research and conduct UAV field tests. In particular, the recent trial results
released by Qualcomm have shown that the current fourth-generation (4G) cellular network can
provide reliable communications for UAVs at an altitude up to 400 feet [5], which paves the
way for realizing cellular-enabled UAV communications in future. Besides being “drone clients”
of wireless networks, UAVs can also be employed as various aerial communication platforms,
to help improve the performance of existing wireless communication systems such as cellular
networks [6].
Compared to the traditional terrestrial wireless communications, UAV-enabled communica-
tions mainly have the following three advantages. First, they are significantly less affected by
channel impairments such as shadowing and fading, and in general possess more reliable air-
to-ground channels due to the much higher possibility of having line-of-sight (LoS) links with
ground users. Second, UAVs can be deployed more flexibly and moved more freely in the
three-dimensional (3D) space. Third, the high mobility of UAVs can be fully controlled to
enhance desired communication links and/or avoid undesired interference via proper trajectory
design. These features bring both opportunities and challenges in designing UAV-enabled wireless
communications, which were not explored before in conventional terrestrial systems with fixed
ground base stations (BSs) [6]. As UAVs are suitable to serve as airborne communication hubs3 such as aerial BSs and/or relays, they are especially useful for the practical scenarios that require on-demand deployment in temporary events or emergency situations (such as natural disaster), when the ground infrastructures are insufficient or even unavailable. However, there are also open challenges that need to be tackled before realizing the full potential of UAV-enabled communications, such as UAV deployment and trajectory design, communications resource allocation and multiple access, etc. Recently, UAV deployment problems have been extensively studied in the literature [7]–[12]. The main design objective is to optimize the UAVs’ altitude, horizontal positions, and/or spatial density to achieve the maximum communication coverage in a given area. However, the high mobility or trajectory design of UAVs is not considered in these works. An energy-efficient relaying scheme is proposed in [13] where multiple UAVs cooperatively relay data packets from ground sensors to a remote BS based on time division multiple access (TDMA). Although the UAV mobility is considered in [13], the UAVs’ trajectories are assumed to be pre-determined and not optimized, which simplifies the design to a UAV-packet matching problem. In [14], both the static and mobile UAV-enabled wireless networks are studied which are underlaid with a device-to-device (D2D) communication network. Yet, the mobile UAV is only allowed to communicate at a set of stop points. As a result, the UAV trajectory is highly restricted and does not fully exploit the UAV’s high mobility for performance optimization. As such, a joint UAV trajectory and adaptive communication design is more promising. Motivated by this, a general trajectory and communication joint optimization framework is proposed in [15] for a UAV- enabled mobile relaying system, which is also extended to the energy efficiency maximization in a point-to-point UAV-ground communication system [16]. The UAV trajectory designs in [15] and [16] can be considered as a generalization of the UAV deployment problem, subject to practical constraints on the UAV’s mobility, such as its initial/final locations, maximum speed and acceleration, etc. Thus, it is intuitive that a mobile UAV with optimized trajectory in general can achieve higher throughput than a static UAV, even with optimal deployment/placement. For UAV-enabled multiuser communication networks, a novel cyclical multiple access scheme is proposed in [17] where the UAV periodically serves each of the ground users along its cyclical trajectory via TDMA. In [18], a joint user scheduling, power control, and trajectory optimization problem is investigated for a multi-UAV enabled multiuser system. It is shown that with joint trajectory design and power control, the UAVs can cooperatively serve the ground users in a periodic manner with strong LoS links and yet avoiding severe interference. However, this is
4 achieved by scheduling each user to communicate only when its associated UAV is sufficiently close to it in each UAV flight period, which implies that the user has to wait for the next UAV flight period to communicate again. As a result, the throughput gain brought by the UAV’s mobility does not come for free but in fact at the cost of user communication delay. Future wireless networks are expected to provide different quality-of-service (QoS) guarantees for a wide range of applications with diversified requirements [19], [20]. In fact, the delay requirements of wireless multimedia services may vary dramatically in a large scale from milliseconds such as for video conferencing and online gaming, to several seconds for file downloading/sharing and data backup. For example, when a user is downloading files which are delay-tolerant in general, some other users in the same area may be watching high-definition (HD) movies on YouTube that require minimum rates at any time. In light of this, wireless networks with such mixed services should be optimized to not only maximize the system total throughput but also meet the heterogeneous delay requirements of different applications. Although the UAV trajectory design has been shown to significantly enhance the throughput of various wireless communication systems such as for mobile relaying channel [13], [15], multiple access channel (MAC) and broadcast channel (BC) [17], [21], [22], interference channel (IFC) [14], [18], and wiretap channel [14], [23], all these works consider only delay-tolerant applications where the users’ instantaneous rates in general cannot be guaranteed. Therefore, it remains unknown whether mobile UAVs are still able to provide throughput gains over static UAVs/ground BSs, when the users’ delay or minimum-rate requirements are considered. Motivated by such an open question, we study in this paper a UAV-enabled orthogonal frequency division multiple access (OFDMA) system, where a UAV is dispatched as a mobile BS to serve a group of users on the ground during a given finite period, as shown in Fig. 1. Besides having been standardized as the downlink multiple access scheme in the current 4G networks, OFDMA is also deemed as a promising candidate for the forthcoming fifth-generation (5G) wireless networks [24]–[26]. This is essentially attributed to its various advantages, e.g., flexible bandwidth and power allocation over users, which fits particularly well in our considered UAV-enabled network with heterogeneous user delay requirements. Without loss of generality, we consider two types of data traffic for each user: delay-constrained traffic where a minimum rate should be supported at any time during the UAV service period versus delay-tolerant traffic where no minimum rate is required and the transmission rate can be elastically allocated over the period. To this end, we introduce a minimum-rate ratio (MRR) for each user which is defined as
5 Fig. 1. A UAV-enabled wireless network. the minimum instantaneous rate required over the achievable average throughput of the whole period. As such, the MRR can be flexibly adjusted by each user to specify the percentage of the delay-constrained traffic required versus that of the delay-tolerant traffic, depending on real-time applications. Our goal is to maximize the minimum average throughput of all ground users while meeting a given set of constraints on the users’ MRRs, by jointly optimizing the UAV trajectory and OFDMA resource allocation. Compared to prior works [13]–[18], [21], [27], [28], such a joint UAV trajectory and resource allocation design is more general and practically useful since the user communication delay requirements are taken into account. This thus leads to a more fair throughput comparison with the terrestrial BSs or static UAVs and also helps to reveal a better understanding of the fundamental throughput gain achievable by exploiting the UAV’s mobility control subject to delay constraints. Intuitively, when the UAV flies towards some users to capture better channels with them, it gets farther away spontaneously from other users that are not in its heading direction and thus experiences degraded channels. As a result, more bandwidth and transmit power need to be allocated to those users so that their minimum rates can be achieved. This in turn would limit the potential rate increase of the users in the UAV’s heading direction. Thus, there exists a new and non-trivial tradeoff in the UAV trajectory design for throughput maximization when minimum-rate or delay constraints are considered. The main contributions of this paper are summarized as follows. First, we formulate a joint UAV trajectory and OFDMA resource allocation optimization problem to maximize the minimum
6 throughput of ground users while guaranteeing their specified MRR constraints. Next, we show that the system max-min throughput is non-increasing with respect to the users’ MRRs, which implies that the throughput gain of mobile UAVs over static UAVs reduces as the delay constraints become more stringent. Although the formulated problem is non-convex and challenging to solve, we propose an efficient iterative block coordinate descent algorithm to solve the bandwidth and power allocation problem and the UAV trajectory optimization problem alternately, which is guaranteed to converge. Specifically, in each iteration, the bandwidth and power allocation problem is solved optimally by applying the Lagrange duality method with given fixed UAV trajectory. While for the UAV trajectory optimization problem with fixed OFDMA resource allocation, the successive convex optimization technique is applied to tackle its non-convexity. However, due to the MRR constraints, it is shown that the conventional block coordinate descent method which directly iterates between the OFDMA resource allocation and UAV trajectory design will very likely get stuck at the initial point, which leads to an ineffective update of the UAV trajectory. To overcome this issue, we propose a new parameter-assisted block coordinate descent method where each parameter is a temporary MRR set to be larger than the target MRR for a corresponding user. Then at each iteration, we gradually decrease the temporary MRRs before solving the UAV trajectory optimization problem, until they reach the target MRRs for all users. It is shown that this new method can effectively update the UAV trajectory in each iteration, thus resolving the issue of conventional block coordinate descent method. Furthermore, we propose a systematic and low-complexity UAV trajectory initialization scheme based on the simple circular trajectory. Finally, numerical results are provided to verify the fundamental tradeoff between the system max-min throughput and the users’ delay/MRR constraints and demonstrate the effectiveness of our proposed designs. The rest of this paper is organized as follows. Section II introduces the system model and the problem formulation for a UAV-enabled OFDMA network. In Section III, we propose an efficient iterative algorithm as well as a general UAV trajectory initialization scheme for the considered problem. Section VI presents the numerical results to demonstrate the performance of the proposed designs. Finally, we conclude the paper in Section VI. Notations: In this paper, scalars are denoted by italic letters, while vectors and matrices are respectively denoted by bold-face lower-case and upper-case letters. RM ×1 denotes the space of M-dimensional real-valued vector. For a vector a, kak represents its Euclidean norm, aT denotes its transpose, and a 0 indicates that a is element-wise larger than or equal to 0. For
7
a time-dependent function x(t), ẋ(t) denotes its derivative with respect to time t. For a set K,
|K| denotes its cardinality.
II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
A. System Model
As shown in Fig. 1, we consider a UAV-enabled OFDMA system where the UAV is employed
as an aerial BS to serve a group of K users on the ground. The user set is denoted by K with
|K| = K. In practice, the users that need to be served by the UAV can be either the terrestrial
BSs that have no ground backhaul links or the ground mobile terminals in a geographical area
that is poorly or not even covered by existing terrestrial BSs. At any time during the UAV flight
period, denoted by T , the UAV can communicate with multiple ground users simultaneously by
employing OFDMA, i.e., assigning each user a fraction of the total bandwidth/transmit power. In
general, from the perspective of throughput maximization, it is intuitive that larger T is desirable
since it will provide the UAV more time to fly closer to each ground user, leading to better air-
to-ground links. The effect of T on the system performance will be investigated in detail in
Section IV. Since we target for a centralized design that is implemented off-line, the proposed
algorithm can be performed at a central controller (e.g., installed on the UAV) who is able to
collect all the users’ information such as their locations and MRRs. Then the obtained solutions
can be programmed into the control and communication units of the UAV.
We assume that the horizontal coordinate of each ground user is known in advance and fixed
at wk = [xk , yk ]T ∈ R2×1 , k ∈ K. The UAV is assumed to fly at a fixed altitude H above
ground and the time-varying horizontal coordinate of the UAV at time instant t is denoted by
q(t) = [x(t), y(t)]T ∈ R2×1 , with 0 ≤ t ≤ T . To serve ground users periodically, we assume that
the UAV needs to return to its initial location by the end of each period T , i.e., q(0) = q(T ).
In addition, the UAV trajectory is also subject to the maximum speed constraints in practice,
i.e, ||q̇(t)|| ≤ Vmax , 0 ≤ t ≤ T , where Vmax denotes the maximum UAV speed in meter/second
(m/s). However, the continuous variable t essentially implies an infinite number of UAV speed
constraints which are difficult to tackle in general. To facilitate our analysis and algorithm design,
we apply the discrete linear state-space approximation technique, which results in a finite number
of constraints. Specifically, we discretize the UAV flight period T into N equally-spaced time
slots with step size δt , i.e., t = nδt , n ∈ N = {1, ..., N}. Note that for the given maximum
UAV speed Vmax and altitude H, the number of time slots N can be chosen sufficiently large8
such that the UAV’s location change within each time slot δt can be assumed to be negligible,
compared to the link distances from the UAV to all ground users. However, a larger value of N
inevitably increases the complexity of the proposed design. Thus, the number of time slots N
can be properly chosen in practice to balance between the accuracy and algorithm complexity.
More discussions on the choice of N as well as its impact on the system performance can be
found in [18]. Based on such a discretization, the UAV trajectory can be characterized by a
sequence of UAV locations q[n] = [x[n], y[n]]T , n = 1, · · · , N. As a result, the above constraints
can be equivalently modeled as
q[1] = q[N], (1)
||q[n + 1] − q[n]||2 ≤ Smax
2
, n = 1, ..., N − 1, (2)
where Smax , Vmax δt is the maximum horizontal distance that the UAV can travel within one
time slot. Furthermore, the distance from the UAV to user k in time slot n is assumed to be a
constant that can be expressed as
p
dk [n] = H 2 + ||q[n] − wk ||2. (3)
The measurement results in [29]–[31] have shown that the air-to-ground communication
channels are mainly dominated by the LoS links even when the UAV is at a moderate altitude.
For example, for the UAV at an altitude of H = 120 m, the LoS probability of air-to-ground
links in rural environment exceeds 95% for a horizontal ground distance of 4 kilometers (Km)
[29]. Thus, we assume that the channel quality depends mainly on the UAV-user distance for
simplicity. In addition, the Doppler effect induced by the UAV mobility is assumed to be perfectly
compensated at the receivers. Following the free-space path loss model, the channel power gain
from the UAV to user k in time slot n can thus be expressed as
ρ0
hk [n] = ρ0 d−2
k [n] = , (4)
H 2 + ||q[n] − wk ||2
where ρ0 denotes the channel power gain at the reference distance d0 = 1 m. Denote the total
available system bandwidth by B in Hertz (Hz). The fraction of bandwidth assigned to user k
in time slot n is denoted by αk [n]. In a practical OFDMA system, αk [n] is in general a discrete
value between 0 and 1, which increases linearly with the number of subcarriers assigned to user
k in time slot n. It is known that when the number of subcarriers is sufficiently large, αk [n]9
can be approximated to a continuous value between 0 and 1. Thus, the bandwidth allocation
constraint can be expressed as
K
X
αk [n] ≤ 1, ∀ n, (5)
k=1
0 ≤ αk [n] ≤ 1, ∀ k, n. (6)
Denote the transmit power allocated to user k in time slot n by pk [n] ≥ 0. Then the total transmit
power constraint of the UAV can be expressed as
K
X
pk [n] ≤ Pmax , ∀ n, (7)
k=1
where Pmax is the maximum allowed transmit power of the UAV in each time slot. Accordingly,
the instantaneous achievable rate of user k in time slot n, denoted by rk [n] in bits/second/Hz
(bps/Hz), can be expressed as
pk [n]hk [n]
rk [n] = αk [n] log2 1 +
αk [n]BN0
pk [n]γ0
= αk [n] log2 1 + , (8)
αk [n](H 2 + ||q[n] − wk ||2 )
ρ0
where γ0 , BN0
, with N0 denoting the power spectral density of the additive white Gaussian
noise (AWGN) at the receivers. As a result, the average achievable throughput of user k over
N time slots in bps/Hz, is given by
N
1 X
Rk = rk [n]. (9)
N n=1
Motivated by the diversified user applications and heterogeneous delay requirements in the
forthcoming 5G wireless networks, we consider both delay-constrained and delay-tolerant ser-
vices for users in the system. Specifically, a parameter θk , θk ∈ [0 1], is introduced to denote the
MRR of user k, which means that at any time slot, θk fraction of its average throughput over
N slots is delay-constrained and the remaining 1 − θk fraction is delay-tolerant. In particular,
θk = 0 and θk = 1 indicate that all the services of user k are delay-tolerant and delay-constrained,
respectively. As such, the MRR constraint of user k in time slot n can be expressed as
rk [n] ≥ θk Rk , ∀ k, n, (10)
which implies that in any of the N time slots, at least θk fraction of the average throughput
needs to be satisfied for each user k. Therefore, a system where some users’ services are all10
delay-constrained while those of the others are all delay-tolerant, i.e., θk ∈ {0, 1}, ∀ k, is a special
case of our general setup.
B. Problem Formulation
Let A = {αk [n], k ∈ K, n ∈ N }, P = {pk [n], k ∈ K, n ∈ N }, and Q = {q[n], n ∈ N }. By
taking into account the MRR constraints of all users, we aim to maximize the minimum average
throughput among them via jointly optimizing the bandwidth and power allocation (i.e., A and
P) as well as the UAV trajectory (i.e., Q). Define η , min Rk . The optimization problem is
k∈K
formulated as
max η (11a)
η,A,P,Q
s.t. Rk ≥ η, ∀ k, (11b)
rk [n] ≥ θk Rk , ∀ k, n, (11c)
K
X
pk [n] ≤ Pmax , ∀ n, (11d)
k=1
pk [n] ≥ 0, ∀ k, n, (11e)
K
X
αk [n] ≤ 1, ∀ n, (11f)
k=1
0 ≤ αk [n] ≤ 1, ∀ k, n, (11g)
||q[n + 1] − q[n]||2 ≤ Smax
2
, n = 1, ..., N − 1, (11h)
q[1] = q[N]. (11i)
Note that the challenges of solving problem (11) lie in the following three aspects. First, Rk
and rk [n] in constraints (11b) and (11c) are not jointly concave with respect to the optimization
variables A, P, and Q. Second, for fixed UAV trajectory Q, although Rk and rk [n] are jointly
concave with respect to A and P, (11c) is non-convex due to the presence of Rk in its left-hand-
side (LHS). Third, for fixed bandwidth and power allocation A and P, Rk and rk [n] are neither
convex nor concave with respect to Q. Consequently, problem (11) is a non-convex optimization
problem and in general, there is no standard method for solving such a problem efficiently.11
To tackle the above challenges, we first transform problem (11) into a more tractable form as
follows,
max η (12a)
η,A,P,Q
s.t. rk [n] ≥ θk η, ∀ k, n, (12b)
(11b), (11d), (11e), (11f), (11g), (11h), (11i). (12c)
Comparing (12b) with (11c), it follows that the feasible set of problem (11) is a subset of that
of problem (12) in general. However, the equivalence of problems (11) and (12) holds if all
users achieve the equal average throughput in the optimal solution to problem (11). This can be
easily verified since otherwise the objective value of (11) can be further improved by allocating
more transmit power and/or bandwidth to the user with a lower average throughput without
violating the total transmit power and bandwidth allocation constraints (11d) and (11f). With
such a transformation, we only need to focus on solving problem (12) in the rest of the paper.
Although it is still a non-convex optimization problem, problem (12) facilitates the development
of an efficient algorithm. Before proceeding to solve problem (12), we first show the effect of
the users’ MRRs on the maximum objective value of problem (12).
Theorem 1. The maximum objective value of problem (12) is an element-wise non-increasing
function with respect to {θk }.
Proof. Denote the optimal solutions of problem (12) with θ ∗ = {θk∗ , k ∈ K} and θ̂ = {θ̂k , k ∈ K}
by S ∗ = {η ∗ , αk∗ [n], p∗k [n], q∗ [n], k ∈ K, n ∈ N } and Ŝ = {η̂, α̂k [n], p̂k [n], q̂[n], k ∈ K, n ∈ N },
respectively. To prove Theorem 1, we only need to show that η ∗ ≤ η̂ holds when θ̂ θ ∗ , where
indicates element-wisely less than or equal to, i.e., θ̂k ≤ θk∗ , ∀ k. Note that in problem (12),
the MRRs are only involved in constraint (12b). Thus, we have the following inequalities
∗ ∗ p∗k [n]γ0
rk [n] = αk [n] log2 1 + ∗ ≥ θk∗ η ∗ ≥ θ̂k η ∗ , ∀ k, n, (13)
αk [n](H 2 + ||q∗ [n] − wk ||2 )
which implies that S ∗ is also a feasible solution of problem (12) with θ̂. Since η̂ is the maximum
objective value of problem (12) with θ̂, it follows that η ∗ ≤ η̂, which thus completes the proof
of Theorem 1.
Theorem 1 sheds light on the fundamental tradeoff between the max-min average throughput
and the user communication delay requirement: as the required MRR θk increases for any user12
k, the max-min average throughput of the system decreases in general. This is because imposing
more stringent minimum-rate requirements on users fundamentally limit the UAV’s mobility to
fly closer to any user to achieve a better channel since at the same time it needs to meet the
minimum-rate requirements of other users that are not in its heading direction and thus will
have degraded channels with it. As a result, the degree of freedom for exploiting the UAV’s
mobility via its trajectory design is restricted by such delay/MRR constraints, thus leading to
lower max-min average throughput.
III. P ROPOSED S OLUTION
In this section, we propose an efficient parameter-assisted block coordinate descent algorithm
for solving problem (12). Specifically, we first optimize the bandwidth and power allocation
for given UAV trajectory and then optimize the UAV trajectory for given bandwidth and power
allocation. These two optimization problems are solved alternately until convergence is achieved.
A. Joint Bandwidth and Power Allocation
Besides being a subproblem of problem (12), the bandwidth and power allocation optimization
problem for given UAV trajectory may also correspond to a practical scenario when the UAV
trajectory is pre-specified due to other tasks such as aerial imaging, rather than being optimized
for achieving best communication performance. Specifically, for any given UAV trajectory Q,
the bandwidth and power allocation {A, P} can be optimized by solving the following problem,13
max η (14a)
η,A,P
N
1 X pk [n]gk [n]
s.t. αk [n] log2 1 + ≥ η, ∀ k, (14b)
N n=1 αk [n]
pk [n]gk [n]
αk [n] log2 1 + ≥ θk η, ∀ k, n, (14c)
αk [n]
K
X
pk [n] ≤ Pmax , ∀ n, (14d)
k=1
pk [n] ≥ 0, ∀ k, n, (14e)
K
X
αk [n] ≤ 1, ∀ n, (14f)
k=1
0 ≤ αk [n] ≤ 1, ∀ k, n, (14g)
k [n]
where gk [n] , hBN 0
. Define αk [n] log2 1 + pk α[n]g k [n]
k [n]
, 0 when αk [n] = 0, ∀ k, n, such that
the LHSs of both (14b) and (14c) are continuous with respect to αk [n] over the whole domain 0 ≤
αk [n] ≤ 1. As such, problem (14) is a convex optimization problem since αk [n] log2 1 + pk α[n]g k [n]
k [n]
in (14b) and (14c) is jointly concave with respect to αk [n] and pk [n] and (14d)-(14g) are all
affine constraints. Furthermore, it can be verified that Slater’s constraint qualification is satisfied
for problem (14) [32]. Therefore, strong duality holds and the duality gap between problem
(14) and its dual problem is thus zero, which means that the optimal solution can be obtained
efficiently by applying the Lagrange duality. The partial Lagrange function of problem (14) can
be expressed as
L(η, A, P, λ, µ, β, ν)
K N ! N K
!
X 1 X pk [n]gk [n] X X
=η + λk αk [n] log2 1 + −η + βn Pmax − pk [n]
N n=1 αk [n] n=1
k=1 k=1
K X
N X N K
!
X pk [n]gk [n] X
+ µk,n αk [n] log2 1 + − θk η + νn 1 − αk [n]
αk [n]
k=1 n=1 n=1 k=1
K K X N
! K X N
X X X λk pk [n]gk [n]
= 1− λk − µk,n θk η + + µk,n αk [n] log2 1 +
n=1 n=1
N αk [n]
k=1 k=1 k=1
K X
X N K X
X N N
X N
X
− βn pk [n] − νn αk [n] + βn Pmax + νn , (15)
k=1 n=1 k=1 n=1 n=1 n=1
where λ = {λk , ∀ k}, µ = {µk,n, ∀ k, n}, β = {βn , ∀ n}, and ν = {νn , ∀ n} are the non-negative
Lagrange multipliers associated with constraints (14b), (14c), (14d), and (14f), respectively. The14
boundary constraints (14e) and (14g) will be absorbed into the optimal solution in the following.
Accordingly, the dual function is given by
f (λ, µ, β, ν) = max L(η, A, P, λ, µ, β, ν) (16a)
η,A,P
s.t. pk [n] ≥ 0, ∀ k, n, (16b)
0 ≤ αk [n] ≤ 1, ∀ k, n, (16c)
for which the following lemma holds.
Lemma 1. To make f (λ, µ, β, ν) bounded from the above in (16), i.e., f (λ, µ, β, ν) < +∞, it
P PK PN
follows that K
k=1 λk − k=1 n=1 µk,n θk = 1 must hold.
PK PK PN PK
Proof. This is shown by contradiction. If k=1 λk − k=1 n=1 µk,n θk > 1 or k=1 λk −
PK PN
k=1 n=1 µk,n θk < 1, it follows that f (λ, µ, β, ν) → +∞ by setting η → −∞ or η → +∞.
Thus, neither of the above two inequalities can be true and the lemma is proved.
Lemma 1 imposes additional constraints for dual variables λ and µ. As such, the dual problem
of problem (14) is given by
min f (λ, µ, β, ν) (17a)
λ,µ,β,ν
K
X K X
X N
s.t. λk − µk,n θk = 1, (17b)
k=1 k=1 n=1
λ 0, µ 0, β 0, ν 0. (17c)
Next, we show how to obtain the primal optimal solution by applying the Lagrange duality.
1) Obtaining f (λ, µ, β, ν) by Solving Problem (16) : With the given dual variables, problem
(16) can be decomposed into KN + 1 subproblems that can be solved independently in par-
allel. Specifically, one subproblem is for optimizing η and the other KN subproblems are for
optimizing A and P, i.e.,
K K X
N
!
X X
max 1− λk − µk,n θk η, (18)
η
k=1 k=1 n=1
λk pk [n]gk [n]
max + µk,n αk [n] log2 1 + − βn pk [n] − νn αk [n] (19a)
αk [n],pk [n] N αk [n]
s.t. pk [n] ≥ 0, ∀ k, n, (19b)
0 ≤ αk [n] ≤ 1, ∀ k, n, (19c)15
P
where each subproblem in (19) is for user k in time slot n. For problem (18), since K k=1 λk −
PK PN
k=1 n=1 µk,n θk = 1 holds as in Lemma 1, the objective value is zero which is independent of
the value of η. This implies that we can choose any arbitrary real number as the optimal solution,
denoted by η ⋆ . Without loss of generality, we simply set η ⋆ = 0 for the purpose of obtaining the
dual function f (λ, µ, β, ν) and updating the dual variables1. Next, we consider problem (19).
Since problem (19) is jointly concave with respect to pk [n] and αk [n], the solution that satisfies
the Karush-Kuhn-Tucker (KKT) conditions is also the optimal solution. By taking the derivative
of the objective function of (19) with respect to pk [n], the optimal power allocation, denoted by
p⋆k [n], can be obtained as
+
λk + Nµk,n 1
p⋆k [n]
= αk [n] − , (20)
Nβn ln 2 gk [n]
h i+
p⋆ [n] λk +N µk,n 1
where [x]+ , max{x, 0}. Let pek [n] , αkk [n] = N βn ln 2
− gk [n]
, which can be regarded as the
power spectrum density of user k in time slot n. Note that in (20), the power allocation follows
a multi-level water-filling structure. Substituting the obtained p⋆k [n] into problem (19) yields
λk + Nµk,n
max log2 (1 + pek [n]gk [n]) − βn pek [n] − νn αk [n] (21a)
αk [n] N
s.t. 0 ≤ αk [n] ≤ 1, ∀ k, n. (21b)
It is evident that problem (21) is a linear program (LP) with only one optimization variable,
αk [n]. Thus, the optimal bandwidth allocation, denoted by αk⋆ [n], can be obtained as
λk + Nµk,n
1, if log2 (1 + pek [n]gk [n]) − βn pek [n] − νn > 0,
N
⋆
αk [n] = a, λk + Nµk,n (22)
if log2 (1 + pek [n]gk [n]) − βn pek [n] − νn = 0,
N
0, otherwise, ∀ k, n.
where a can be any arbitrary real number between 0 and 1 since the objective value of problem
(21) is not affected in this case. For simplicity, we set a = 0 as for the case of η. In general, (22)
cannot provide the optimal primal solution for problem (14) even with optimal dual variables.
Nevertheless, with the above proposed solutions to problems (18) and (19), the dual function
f (λ, µ, β, ν) is obtained.
1
We note that η ⋆ = 0 cannot be the optimal primal solution to problem (14). How to obtain the optimal primal solution for
this problem will be discussed later in Section III-A-3).16
2) Obtaining Optimal Dual Solution to Problem (17) : After obtaining (η ⋆ , A⋆ , P⋆ ) for given
λ, µ, β, and ν, we next solve the dual problem (17) to find the optimal dual variables that
maximize f (λ, µ, β, ν). Note that although the dual function f (λ, µ, β, ν) is always convex by
definition, it is non-differentiable in general. As a result, the commonly used subgradient based
method such as the ellipsoid method, can be used to solve problem (17). In each iteration, the dual
variables λ, µ, β, and ν are updated based on the subgradients of both the objective function and
the constraint functions in problem (17). Specifically, the subgradient of the objective function
is denoted by s0 = [∆λT , ∆µT , ∆β T , ∆ν T ]T where ∆λ, ∆µ, ∆β, and ∆ν are vectors with the
elements respectively given by
N
1 X pk [n]gk [n]
∆λk = αk [n] log2 1 + , ∀ k, (23)
N n=1 αk [n]
pk [n]gk [n]
∆µk,n = αk [n] log2 1 + , ∀ k, n, (24)
αk [n]
K
X
∆βn = Pmax − pk [n], ∀ n, (25)
k=1
K
X
∆νn = 1 − αk [n], ∀ n. (26)
k=1
Furthermore, the equality constraint in (17b) is equivalent to two inequality constraints: 1 −
PK PK PN PK PK PN
k=1 λ k − k=1 n=1 µ k,n θk ≤ 0 and −1 + k=1 λ k + k=1 n=1 µk,n θk ≤ 0. Thus, the
subgradient of the former constraint function is denoted by s1 = [∆λT , ∆µT , ∆β T , ∆ν T ]T
where the corresponding elements are given by ∆λk = −1, ∆µk,n = −θk , ∆βn = 0, ∆νn = 0. In
addition, the subgradient of the latter constraint function is given by s2 = −s1 . With the above
subgradients, the dual variables can be updated by the constrained ellipsoid method toward the
optimal solution with global convergence [33].
3) Constructing Optimal Primal Solution to Problem (14) : Based on the obtained dual opti-
mal solution λ∗ , µ∗ , β ∗ , and ν ∗ , it remains to obtain the optimal primal solution {η ∗ , A∗, P∗ } to
problem (14). It is worth pointing out that for a convex optimization problem, the optimal solution
that maximizes the Lagrangian function is the optimal primal solution if and only if such a solu-
tion is unique and primal feasible [32]. However, in our case, the optimal solutions η ⋆ and A⋆ that
P PK PN
maximize L(η, A, P, λ, µ, β, ν) are not unique given that K k=1 λk − k=1 n=1 µk,n θk = 1 and
λk +N µk,n
N
log2 (1 + pek [n]gk [n]) − βn pek [n] − µn = 0 as in (18) and (22). Therefore, additional
steps are needed in order to construct the optimal primal solution. The key observation is that17
Algorithm 1 Joint bandwidth and power allocation algorithm for solving problem (14).
1: Initialize λ, µ, β, ν, and the ellipsoid.
2: repeat
3: Solve problem (18) and (19) to obtain η ⋆ , A⋆ , and P⋆ .
4: Compute the subgradients of the objective function and the constraint functions in problem
(17).
5: Update λ, µ, β, and ν by using the constrained ellipsoid method.
6: until λ, µ, β, and ν converge within a prescribed accuracy.
7: Set (λ∗ , µ∗ , β ∗ , ν ∗ ) ← (λ, µ, β, ν).
8: Obtain η ∗ , A∗ , and P∗ by solving problem (27) and using (20).
with given λ∗ , µ∗ , and β ∗ , the optimal power spectrum density (the ratio of the optimal power
p∗k [n]
allocation to the optimal bandwidth allocation), i.e., pe∗k [n] = α∗k [n]
, can be uniquely obtained
from (20). By substituting pe∗k [n] into the primal problem (14), we have
max η (27a)
η,A
N
1 X
s.t. αk [n] log2 (1 + pe∗k [n]gk [n]) ≥ η, ∀ k, (27b)
N n=1
αk [n] log2 (1 + pe∗k [n]gk [n]) ≥ θk η, ∀ k, n, (27c)
K
X
p∗k [n] ≤ Pmax , ∀ n,
αk [n]e (27d)
k=1
K
X
αk [n] ≤ 1, ∀ n, (27e)
k=1
0 ≤ αk [n] ≤ 1, ∀ k, n. (27f)
Given pe∗k [n], it is easy to observe that problem (27) is an LP with respect to A and η, which
thus can be efficiently solved by using standard convex optimization solvers such as CVX [34].
After obtaining the optimal bandwidth allocation A∗ , the corresponding power allocation P∗
can be obtained as p∗k [n] = pe∗k [n]αk∗ [n], ∀ k, n. The details of the procedures for obtaining the
optimal solution to problem (14) are summarized in Algorithm 1. The computational complexity
of Algorithm 1 consists of three parts. The first part is for solving problems (18) and (19), the
second part is for updating the dual variables by the ellipsoid method, and the third part is18
for solving linear program problem (27). In step 3) of Algorithm 1, the complexity of solving
problem (18) is O(1) and that of solving (19) is O(KN). The complexities of step 4) and 5) are
O(KN) and O(K 2N 2 ), respectively. Since the ellipsoid method takes O(K 2 N 2 ) to converge, the
total complexity for step 2) to 6) is O(K 4 N 4 ) [33]. The complexity of solving (27) is O(K 3 N 3 ).
Therefore, the total complexity of Algorithm 1 is O(K 4N 4 ).
B. UAV Trajectory Optimization
Given any feasible bandwidth and power allocation {A, P}, problem (12) is simplified into
the following problem for optimizing the UAV trajectory Q only, i.e.,
max η (28a)
η,Q
N
1 X γk [n]
s.t. αk [n] log2 1 + 2 ≥ η, ∀ k, (28b)
N n=1 H + ||q[n] − wk ||2
γk [n]
αk [n] log2 1 + 2 ≥ θk η, ∀ k, n, (28c)
H + ||q[n] − wk ||2
||q[n + 1] − q[n]||2 ≤ Smax
2
, n = 1, ..., N − 1, (28d)
q[1] = q[N], (28e)
pk [n]ρ0
where γk [n] , αk [n]BN0
. Note that problem (28) is not a convex optimization problem since the
LHSs of constraints (28b) and (28c) are not concave with respect to q[n]. In general, there is
no efficient method to obtain the optimal solution for such a non-convex problem. However,
we observe that the LHSs of both (28b) and (28c) are convex with respect to ||q[n] − wk ||2 .
Note that for a convex function, its first-order Taylor expansion is the global under-estimator at
any point [32]. This thus motivates us to leverage the successive convex optimization technique
to tackle the non-convex problem (28) by an iterative algorithm, where in each iteration, the
LHSs of both (28b) and (28c) are replaced by more tractable functions derived from the Taylor
expansion at a given local point. Specifically, with given local point qr [n], we have the following
inequality
γ0
rk [n] = αk [n] log2 1 +
H + ||q[n] − wk ||2
2
≥ αk [n] −Ark [n] ||q[n] − wk ||2 − ||qr [n] − wk ||2 + Bkr [n]
, rklb,r [n], (29)19
where
γ0 log2 e
Ark [n] = , (30)
(H 2
+ ||qr [n] − wk || )(H 2 + ||qr [n]
2 − wk ||2 + γ0 )
γ0
Bkr [n] = log2 1 + , ∀ k, n. (31)
H + ||q [n] − wk ||2
2 r
PN lb,r
For any given local point Qr , define the function η lb,r (A, Q) = min n=1 rk [n]. With
i∈K
the lower bounds rklb,r [n], ∀ k, in (29) and Qr , problem (28) is approximated as the following
problem
max η lb,r
ηlb,r ,Q
N
1 X lb,r
s.t. r [n] ≥ η lb,r , ∀ k, (32a)
N n=1 k
rklb,r [n] ≥ θk η lb,r , ∀ k, n, (32b)
||q[n + 1] − q[n]||2 ≤ Smax
2
, n = 1, ..., N − 1, (32c)
q[1] = q[N]. (32d)
Note that constraints (32a), (32b), and (32c) are all convex quadratic constraints and (32d) is a
linear constraint. Therefore, problem (32) is a convex quadratically constrained quadratic program
(QCQP) that can be solved within a polynomial complexity by standard convex optimization
solvers such as CVX [34]. It is worth pointing out that due to the lower bounds adopted from
(29), constraints (32a) and (32b) imply (28b) and (28c), respectively, but the reverse does not
hold in general. In this regard, the optimal objective value obtained by solving problem (32)
always serves as a lower bound for that of problem (28).
C. Overall Algorithm Design
In this section, we propose an efficient iterative algorithm to solve problem (12) based on the
results in previous two sections. Note that the conventional block coordinate descent method
alternately optimizes one block of optimization variables in each iteration while keeping other
blocks of optimization variables fixed, until the convergence is achieved [35], [36]. However,
directly applying such a block coordinate descent method for our considered problem, i.e., by
alternately optimizing one block from the power and bandwidth allocation variables {A, P} in
problem (14) and the UAV trajectory variables Q in problem (28), with the other block fixed,20
may fail to update the UAV trajectory effectively. This can be observed from the MRR constraints
in problem (28), i.e.,
γk [n]
αk [n] log2 1 + ≥ θk η, ∀ k, n. (33)
H + ||q[n] − wk ||2
2
For given αk [n] and pk [n], ∀ k, n, since problem (28) aims to increase η by optimizing UAV
trajectory q[n], the right-hand-side (RHS) of (33) is expected to increase after each iteration. As
a result, for users that have met constraints in (14c) with equality in the last iteration, the only
way to increase the LHS of (33) in the current iteration is by decreasing ||q[n] − wk ||2 , ∀ k. This
implies that in each time slot n, the UAV’s location q[n] needs to be updated to decrease the
distances from the UAV to all these users. Thus, the freedom for optimizing the UAV trajectory
is severely limited which can lead to ineffective update of the UAV trajectory in each iteration.
To tackle this issue, we propose a new parameter-assisted block coordinate descent method
where each parameter is a temporary MRR for a corresponding user k, denoted by θtemp,k , which
is set larger than the actual MRR target θk , if 0 < θk < 1. The main idea is to use the newly
introduced temporary MRR θtemp,k > θk for solving problem (28) rather than directly using the
target θk . Specifically, θtemp,k for each user k is gradually decreased before solving problem (28)
in each iteration, until the target MRR θk is achieved for all users. As such, η will increase
after each iteration, while the constraints in (33) will be relaxed due to the decrease of θtemp,k ’s,
which thus permits a more effective UAV trajectory update in each iteration compared to the
conventional block coordinate decent method. Furthermore, the proposed method also generates
a feasible solution for the original problem (12) since θk will be eventually achieved by gradually
decreasing θtemp,k with a predefined step size, θstep,k > 0. The details of the proposed method
are summarized in Algorithm 2.
The convergence of the proposed Algorithm 2 can be analyzed as follows. From step 4, it
can be seen that θk , ∀ k, is non-increasing over the iterations. Recall that the objective value of
problem (12) is element-wise non-increasing with respect to θk , ∀ k. Thus, it can be shown that
the objective value achieved by Algorithm 2 is non-decreasing in each iteration. Furthermore,
the objective value of problem (12) is bounded from the above. Thus, the proposed Algorithm
2 is guaranteed to converge and the obtained solution is also feasible for the original problem
(11). Although the obtained solution is generally suboptimal, we validate the effectiveness of
Algorithm 2 via simulations by comparing with other benchmark schemes in Section IV.21
Algorithm 2 Parameter-assisted block coordinate descent algorithm for solving problem (12).
θini,k −θk
1: Initialize Q0 , θini,k , and Lmax . Let r = 0, θtemp,k = θini,k , and θstep,k = Lmax
, ∀ k.
2: repeat
3: Solve problem (14) by applying Algorithm 1 for given Qr , and denote the optimal solution
as {Ar+1, Pr+1 }.
4: θtemp,k = max{θtemp,k − (r + 1)θstep,k , θk }, ∀ k.
5: Solve problem (28) for given {Ar+1, Pr+1 , Qr , θtemp,k }, and denote the optimal solution
as {Qr+1 }.
6: Update r = r + 1.
7: until θtemp,k = θk , ∀ k, and the fractional increase of the objective value is below a threshold
ǫ > 0.
D. UAV Initial Trajectory and Users’ MRR Initialization
In Algorithm 2, the initial UAV trajectory, Q0 = {q0 [n], ∀ n}, and the initial temporary users’
MRRs, θini,k , ∀ k, need to be set. First, we propose in the following a general and systematic
trajectory initialization scheme based on the simple circular UAV trajectory, which also includes
the one proposed in [21] with θk = 0, ∀ k, as a special case. For a circular UAV trajectory, we
need to obtain the circle center cini = [xini , yini]T and radius rini . For convenience, the circle
PK
wk
center is set as the geometry center of all users, which is given by cini = k=1
K
. The main idea
for determining the circle radius of the initial circular trajectory is based on the following three
observations. First, it is intuitive that as θk increases, the area covered by the UAV trajectory
shall decrease, as implied in Theorem 1. As such, the initial circle radius rini should decrease
with θk , ∀ k. Second, when θk = 0, ∀ k, the initial circle radius should be simplified to that for
the case when the MRR constraints are not considered as in [21]. Third, it is also known that
when θk = 1, ∀ k, the UAV mobility does not provide any performance gain, which implies that
the UAV should remain static. In such a case, the UAV trajectory becomes a point and the initial
circle radius is thus zero. Based on the above observations, we propose to set the initial circle
radius as
PK !
k=1 θk
rini = 1− r0 , (34)
K
where r0 is the circle radius for the case when θk = 0, ∀ k, or no MRR constraints are considered
[21]. According to [21], r0 is obtained as follows: 1) by using the user geometry center cini as the22
circle center, we first calculate the radius of a minimum circle that can cover all ground users,
denoted by rmin = max ||wk −cini ||; 2) to balance the number of users outside and inside the UAV
k∈K
T rmin
trajectory circle as well as satisfying the UAV speed constraint, we set r0 = min{ Vmax
2π
, 2 }.
From (34), it is easy to check that when θk = 0, ∀ k, rini = r0 and when θk = 1, ∀ k, rini = 0,
as expected. Let ϕn = 2π Nn−1
−1
, ∀ n. With cini and rini , the initial circular UAV trajectory Q0 is
given as follows,
q0 [n] = [xini + rini cos ϕn , yini + rini sin ϕn ]T , n = 1, ..., N. (35)
Next, we initialize the users’ MRRs, θini,k , ∀ k. Note that if user k has no MRR constraint,
i.e., θk = 0, then there exists no constraint for user k in (33). Thus, user k will not cause the
ineffective UAV trajectory update problem previously described in Section III-C and its initial
MRR can be directly set as θini,k = 0. In contrast, as long as user k has a non-trivial MRR
target, i.e., θk > 0, N corresponding MRR constraints in (33) will be imposed for user k which
may cause ineffective updates of the UAV trajectory. Thus, the initial MRRs of users are set as
1, if θk > 0,
θini,k = (36)
0, if θk = 0.
IV. N UMERICAL R ESULTS
In this section, numerical results are provided to validate the proposed joint OFDMA resource
allocation and UAV trajectory design as well as the fundamental throughput-delay tradeoff in
UAV-enabled wireless communications. We consider a system with K = 4 ground users that are
located in a horizontal plane as shown in Fig. 2, marked by ‘×’s. The UAV is assumed to fly
at a fixed altitude H = 500 m [31]. The total available communication bandwidth is B = 10
MHz and the noise power spectrum density at the ground users is assumed to be identical and
set as N0 = −169 dBm/Hz. The channel power gain at the reference distance d0 = 1 m is
set as ρ0 = −50 dB. Other parameters are set as Pmax = 0.1 W, Vmax = 50 m/s, T = 270 s,
and N = 540, respectively, if not specified otherwise. For illustration, all the trajectories in the
simulations are sampled every 4 s and the sampled points are marked by ‘△’s.
A. UAV Trajectory and Max-min Throughput versus Homogeneous MRRs
We first consider the homogeneous delay requirement case when all ground users have the
same MRR, i.e., θk = θ, ∀ k. In Figs. 2 and 3, the UAV trajectory and the max-min throughput are23
1500 1500
1000 1000
500 500
y(m)
y(m)
0 0
−500 −500
−1000 −1000
−1500 −1500
−1500 −1000 −500 0 500 1000 1500 −1500 −1000 −500 0 500 1000 1500
x(m) x(m)
(a) θ = 0 (b) θ = 0.6
1500 1500
1000 1000
500 500
y(m)
y(m)
0 0
−500 −500
−1000 −1000
−1500 −1500
−1500 −1000 −500 0 500 1000 1500 −1500 −1000 −500 0 500 1000 1500
x(m) x(m)
(c) θ = 0.8 (d) θ = 1
Fig. 2. UAV trajectory versus homogeneous MRR θ = θk , ∀ k, for T = 270 s.
illustrated respectively under different MRRs. It can be observed from Fig. 2 that as the MRR,
θ, increases, the UAV’s flight distance decreases and the UAV’s trajectory shrinks gradually
from a square to a smaller ellipse and finally a fixed point, given the same flight period T . In
particular, when θ = 0, i.e., no MRR constraint is considered as in [21], the UAV sequentially
visits and stays above each of the ground users by maximally exploiting its mobility. As such,
the best air-to-ground channel can be realized between the UAV and each ground user. However,
when θ > 0, the UAV’s trajectory adaptively changes depending on the value of θ, as expected.
Notice that in this setup, the closer the UAV flies to one particular user, the farther it is away
from some other users inevitably. As a result, meeting the MRR constraints of these users will
consume more resources (power, bandwidth) and thus becomes the bottleneck for improving the
max-min throughput of the system. Such a situation becomes worse when the MRR and/or the
inter-user distance becomes larger. Generally speaking, with more stringent MRR constraints,
the UAV trajectory tends to be more restricted to avoid getting too far away from any of the
users. Finally, as all ground users have the same MRR constraint as well as the same average24
1
Static UAV
Initial trajectory
0.9
Fly−and−hover trajectory
Proposed trajectory
Max−min throughput (bps/Hz)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0 0.2 0.4 0.6 0.8 1
MRR, θ
Fig. 3. Max-min throughput versus homogeneous MRR θ = θk , ∀ k, for T = 270 s.
throughput, the UAV trajectories are shown to be all symmetric over the users in Fig. 2.
The effect of the MRR constraint on the max-min average throughput is shown in Fig. 3.
Specifically, we compare the following four trajectories: 1) Proposed trajectory which is obtained
by applying Algorithm 2; 2) Fly-and-hover trajectory where the UAV flies with the maximum
speed to visit all the users and hovers (with zero speed) above each of them by equally allocating
the remaining time2; 3) Initial circular trajectory as described in Section III-D; and 4) Static UAV
where the UAV remains static above the geometry center of the users. For all the four schemes
considered, the bandwidth and power allocation is optimized by applying Algorithm 1 with the
corresponding UAV trajectory. First, we observe that the max-min average throughput gradually
decreases with the MRR for the first three considered trajectories with a mobile UAV in general.
This is due to the fundamental tradeoff between the system throughput and the user delay/MRR
constraint, which is in accordance with Theorem 1. Second, when the MRR is small, the proposed
trajectory significantly outperforms the circular trajectory and static UAV. This is because a small
MRR in general implies a large degree of freedom for the UAV trajectory design, but the static
UAV cannot exploit the UAV mobility while the circular trajectory does not fully exploit the
UAV mobility. In contrast, the fly-and-hover trajectory is observed to achieve the throughput very
2
It is worth pointing out that this trajectory is only feasible when T is sufficient large such that the UAV can visit above each
of all ground users.25
1.3
1.2
1.1 θ=0
Max−min throughput (bps/Hz)
θ=0.3
1 θ=0.6
θ=1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0 100 200 300 400 500 600 700 800
Period T (s)
Fig. 4. Max-min throughput versus UAV flight period T .
close to the proposed trajectory for small MRR values. However, when the MRR becomes large,
the fly-and-hover trajectory suffers from a significant throughput loss compared to the proposed
trajectory. This is expected since the fly-and-hover trajectory results in highly asymmetric user
channels over time and thus is inefficient for meeting increasingly more stringent users’ MRR
requirements. Finally, when the MRR reaches the maximum value of one, the max-min average
throughput of the proposed trajectory becomes identical to that of static UAV as the trajectory
converges to the same point as the static UAV.
In Fig. 4, the effect of the UAV flight period T on the max-min average throughput is shown
under different values of the homogeneous MRR, θ. It is observed that the max-min throughput
in the three cases with θ < 1 all increases with T , while for the case of θ = 1, the max-min
throughput remains constant, regardless of T . This suggests that as long as the users have delay-
tolerant data traffic, i.e, θ < 1, the UAV mobility indeed provides throughput gains over a static
UAV with θ = 1 for any T > 0. In addition, such a throughput gain generally increases with
more flight time for the UAV, although the gain is more pronounced when θ is smaller or less
strict delay/MRR constraints are applied. This is because in such cases, as T increases, the UAV
has more time to get closer to each of the users and even be able to hover above them to enjoy
the best channels to them. In contrast, if the services required by users are all delay-constrained,
i.e., θ = 1, then the UAV is unable to achieve any throughput gain over the static case sinceYou can also read
